Decimal representation

In 8.03999\ldots, 9 repeats forever from the third decimal place onward. Since \(0.00999\ldots=0.01\), \(8.03999\ldots=8.03+0.00999\ldots=8.04\). Option 8.039 merely stops the decimal and does not account for the infinitely repeating 9s. Exam tip: When infinitely many 9s follow a digit, increase the relevant preceding digit by 1.

( \frac{29}{40}=0.725 ) and ( \frac{8}{11}=0.7272\ldots ), so (0.7265) lies between them. Convert boundaries into decimals for comparison.

A non-terminating, non-recurring expansion has no repeating block, so it is irrational. A recurring expansion after initial digits is rational. Exam tip: a bar denotes repetition.

(x=2.34375\div100000=0.0000234375). On dividing by (100000), the decimal moves five places left.

The places after the decimal point are tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths and millionths. In 21.005007, the digit 7 is in the sixth place after the decimal point, so its place value is \(\frac{7}{1000000}\). The closest distractor, \(\frac{7}{100000}\), represents the place value of a digit in the fifth decimal place, not the sixth. Exam tip: count the decimal places from left to right carefully.

Write all the numbers up to four decimal places for comparison: \(9.0090, 9.0099, 9.0909, 9.9000\). Comparing the digits after the decimal from left to right gives \(9.0090<9.0099<9.0909<9.9000\). Therefore, option A is correct. Option D incorrectly places \(9.0909\) before \(9.0099\). Exam tip: Add zeros at the end, when needed, to make the decimal places equal before comparing decimals.

(8000\times125=1000000), so ( \frac{137}{8000}=0.017125 ). Convert the denominator into a power of (10) to get the exact decimal.

Find the ratio: 0.03125 ÷ 0.0003125 = 100. Therefore, 0.03125 is 100 times 0.0003125. Equivalently, multiplying 0.0003125 by 100 shifts the decimal point two places to the right and gives 0.03125. Hence, the 10-times option is incorrect; in such questions, divide the compared larger number by the smaller number to find how many times it is greater.

The bar is over 63 only, so the block 63 repeats indefinitely. There are three zeros immediately after the decimal point, followed by 63, 63, 63, ... Hence the correct form is \(0.000636363\ldots\). Option B incorrectly shifts the repeating block one place to the right. Exam tip: write the digits before the bar once, then repeat only the barred block.

Option B is correct. Zero blocks of lengths 1, 2, 3, … prevent a fixed period. Thus the decimal is non-terminating, non-repeating and irrational. Tip: rational decimal expansions terminate or repeat.

Reduce the fraction before examining its decimal expansion. The numerator 84 and denominator 308 have a common factor of 28. Dividing both by 28 gives 84/308 = 3/11. The denominator in lowest form is therefore 11, not 308.

A rational number has a terminating decimal only when the denominator in simplest form contains no prime factors except 2 and 5. Since 11 is another prime factor, the decimal cannot terminate. Indeed, 3/11 = 0.2727..., so the block 27 repeats endlessly. Therefore the decimal expansion is non-terminating recurring, and option C is correct. The fraction’s original denominator does not need to be used after reduction.

The correct value is 11.625. In 11.624999\ldots, 9s continue indefinitely after 4. An infinite repeating tail of 9s can be replaced by increasing the preceding digit by 1; hence 11.624999\ldots = 11.625. The number 11.624 is only a truncated value, so it is not equal to the given decimal. Exam tip: when a decimal ends in \(999\ldots\), add 1 to the digit immediately before that repeating tail.

In 5.070900, the digits after the decimal point represent tenths, hundredths, thousandths, and ten-thousandths in order. The digit 9 is in the fourth decimal place, so its place value is \(9 \times \frac{1}{10000}=\frac{9}{10000}\). Therefore, option C is correct. Exam tip: each place to the right of the decimal point has one-tenth the value of the preceding place.

( \frac{27}{64}=0.421875 ) and (0.421\overline{8}=0.421888\ldots). Therefore, the recurring number is greater.

Write the decimals to four decimal places: \(b=0.5800\), \(a=0.5080\), and \(c=0.5008\). All have 5 in the tenths place. At the hundredths place, \(b\) has 8, so it is the greatest. For \(a\) and \(c\), the hundredths digit is 0, but their thousandths digits are 8 and 0 respectively; hence \(a>c\). Therefore, the descending order is \(b>a>c\). Option \(b>c>a\) incorrectly places \(c\) above \(a\). Exam tip: Add trailing zeroes to make the decimal places equal before comparing decimals.

( \frac{23}{125}=0.184 ) and ( \frac{13}{320}=0.040625 ), so the sum is (0.224625). First convert the fractions into decimals.

( \frac{29}{64}=0.453125 ), so (0.453125-0.265625=0.1875). Convert the fraction into decimal and subtract.

After the decimal point, the first digit is 0 and it occurs only once. Then the block 85 repeats continuously as 85, 85, 85, ... . Hence, the bar is placed only over 85: \(0.0\overline{85}\). \(0.\overline{85}\) is incorrect because it starts repeating 85 immediately after the decimal point. Exam tip: identify the exact repeating block before placing the bar.

On reducing \(\frac{21}{150}\) by 3, we get \(\frac{7}{50}\), and \(50=2\times5^2\). A decimal terminates when the denominator in lowest form has only 2 and 5 as prime factors. Exam tip: reduce first.

The denominator is \\(2^4\times5^7\\). To make a power of 10, the four factors of 2 must be paired with four factors of 5. Three factors of 5 remain, so multiply by \\(2^3\\). The denominator then becomes \\(2^7\times5^7=10^7\\). Thus the terminating decimal can have at most seven decimal places.

Equivalently, the maximum is the larger exponent: \\(\max(4,7)=7\\). Therefore option B is correct. The exponents are not added to give 11, since each matched pair of 2 and 5 forms one factor of 10. Depending on the numerator, the actual decimal may end earlier, but seven is the maximum allowed by this denominator.

The number of zeros between (9)'s increases, so there is no fixed repetition. Such a decimal is non-terminating non-recurring.

To convert a decimal into a percentage, multiply it by 100: \(0.0036 \times 100 = 0.36\). Therefore, the percentage is \(0.36\%\). Option C results from shifting the decimal one place too far. Exam tip: when multiplying a decimal by 100, move the decimal point two places to the right.

The direct answer is option C, 19531.25. Multiplying by 10000000 means multiplying by 10^7. Each multiplication by 10 moves the decimal point one place to the right, so seven zeros move it seven places: 0.001953125 becomes 19531.25. Equivalently, 0.001953125 × 10000000 = 19531.25. Option A, 195.3125, moves the point only five places. Option B, 1953.125, moves it six places. Option C has the required seven-place movement and is correct. Option D, 195312.5, moves the point eight places and is ten times too large. A useful exam cue is: multiplication by 10^n moves the decimal point n places right; add zeros if necessary.

( \frac{23}{32}=0.71875 ), so (0.71825) is greater than (0.718) and less than (0.71875). Convert boundary values into decimals.

In lowest form, a denominator containing any prime factor other than 2 or 5 gives a non-terminating recurring decimal. A denominator with only 2 and 5 gives a terminating decimal. Exam tip: reduce the fraction first.